How to show CES is not an algebra

Question

We say that a subset $V$ of $\{1,2,3,\dots\}$ has Cesaro density $\gamma(V)$ and denote $V\in CES$ if the limit $$\gamma(V):=\lim_{n\to \infty} \frac{\mid V\cap \{1,2,3,\dots, n\}\mid}{n}$$ exists. Give an example of sets $V_1\in CES$ and $V_2\in CES$ for which $V_1\cap V_2\notin CES$.

How to find such an example? If I try to show CES is not algebra.

I can think of something as follows: choose clusters for $V_1\cap V_2$, where in the sequence you go down from, say 0.9 to 0.5 and then up again and so on. This will of course not converge. Between the clusters you have space (where the sequence decreases). Here you can put your elements of $V_1$ and $V_2$ disjointly so that they don't decrease too much. Just an idea. — amsmath, Oct 28 '19 at 01:49
This is basically the same problem formulated using complements: If two sets have a natural (asymptotic) density, does their union? This is also tangentially related: Why is $\mathcal{A}$ not a $\sigma$-algebra? — Martin Sleziak, Dec 07 '19 at 23:22
In fact, here is another one: Do the subsets of $\mathbb N$ that have asymptotic density form an algebra? (It was listed among related questions in the sidebar on the right.) And also this one: Sets with no asymptotical density over $\mathbb N$ — Martin Sleziak, Dec 08 '19 at 11:31

score 7 · Accepted Answer · answered Dec 07 '19 at 22:33

Let $V_1$ denote all odds. Let $V_2 = \{a_1,a_2,\dots\}$ where each $a_i \in \{2i-1,2i\}$. Then each $V_i$ has density $\frac{1}{2}$, but if we have the $a_i$'s start off odd for a while, then $V_1\cap V_2$ will have density $\frac{1}{2}$ for a while, and then if the $a_i$'s be even for a while, the density of $V_1\cap V_2$ will decrease close to $0$, and then we can have the $a_i$'s be odd again for a while, making the density rise close to $\frac{1}{2}$, etc, etc.

[For concreteness, you can have the first $2^{1^2}$ $a_i$'s be odd, then the next $2^{2^2}$ $a_i$'s be even, then the next $2^{3^2}$ $a_i$'s be odd, etc.]

How to show CES is not an algebra

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